This subroutine stores the first n—by default the first ten—Fibonacci numbers in memory, beginning (because, why not?) at address 3867 decimal = F1B hex. Intermediate results are stored in three sequential addresses within the low 256 bytes of memory, which are the most economical to access.

The results are calculated and stored, but are not output to the screen or any other physical device: how to do that would depend on the hardware and the operating system.

This subroutine expects to be called with the value of n{\displaystyle n} in register A, and returns f(n){\displaystyle f(n)} also in A. You may want to take steps to save the previous contents of B, C, and D. The routine only works with fairly small values of n{\displaystyle n}.

/* author: snugsfbay date: March 3, 2016 description: Create a list of x numbers in the Fibonacci sequence. - user may specify the length of the list - enforces a minimum of 2 numbers in the sequence because any fewer is not a sequence - enforces a maximum of 47 because further values are too large for integer data type - Fibonacci sequence always starts with 0 and 1 by definition*/public class FibNumbers{

/* description: method to create a list of numbers in the Fibonacci sequence param: user specified integer representing length of sequence should be 2-47, inclusive. - Sequence starts with 0 and 1 by definition so the minimum length could be as low as 2. - For 48th number in sequence or greater, code would require a Long data type rather than an Integer. return: list of integers in sequence.*/public static List<Integer> makeSeq(Integer len){

if(len<MIN || len==null || len>MAX) { if (len>MAX){ len=MAX; //set length to maximum if user entered too high a value }else{ len=MIN; //set length to minimum if user entered too low a value or none } } //This could be refactored using teneray operator, but we want code coverage to be reflected for each condition

//start with initial list size to find previous two values in the sequence, continue incrementing until list reaches user defined length for(i=fib.size(); i<len; i++){ fib.add(fib[i-1]+fib[i-2]); //create new number based on previous numbers and add that to the list }

Here's what happens:
We replicate the 2-by-2 matrix N times and then apply inner product-replication.
The First removes the shell from the Enclose.
At this point we're basically done, but we need to pick out only Fn{\displaystyle F_{n}} in order to complete the task. Here's one way:

In Babel, we can define fib using a stack-based approach that is not recursive:

fib { <- 0 1 { dup <- + -> swap } -> times zap } <

foo x < puts x in foo. In this case, x is the code list between the curly-braces. This is how you define callable code in Babel. The definition works by initializing the stack with 0, 1. On each iteration of the times loop, the function duplicates the top element. In the first iteration, this gives 0, 1, 1. Then it moves down the stack with the <- operator, giving 0, 1 again. It adds, giving 1. then it moves back up the stack, giving 1, 1. Then it swaps. On the next iteration this gives:

This uses a pre-generated list, requiring much less run-time processor usage. (Since the sequence never changes, this is probably the best way to do this in "the real world". The same applies to other sequences like prime numbers, and numbers like pi and e.)

An unobfuscated iterative implementation.It prints the first N + 1 Fibonacci numbers,where N is taken from standard input.

Ingredients.0 g last1 g this0 g new0 g input

Method.Take input from refrigerator.Put this into 4th mixing bowl.Loop the input.Clean the 3rd mixing bowl.Put last into 3rd mixing bowl.Add this into 3rd mixing bowl.Fold new into 3rd mixing bowl.Clean the 1st mixing bowl.Put this into 1st mixing bowl.Fold last into 1st mixing bowl.Clean the 2nd mixing bowl.Put new into 2nd mixing bowl.Fold this into 2nd mixing bowl.Put new into 4th mixing bowl.Endloop input until looped.Pour contents of the 4th mixing bowl into baking dish.

This is implemented idiomatically as an infinitely long, lazy sequence of all Fibonacci numbers:

(defn fibs [](mapfirst(iterate(fn[[a b]][b (+ a b)])[01])))

Thus to get the nth one:

(nth (fibs)5)

So long as one does not hold onto the head of the sequence, this is unconstrained by length.

The one-line implementation may look confusing at first, but on pulling it apart it actually solves the problem more "directly" than a more explicit looping construct.

(defn fibs [](mapfirst;; throw away the "metadata" (see below) to view just the fib numbers(iterate;; create an infinite sequence of [prev, curr] pairs(fn[[a b]];; to produce the next pair, call this function on the current pair[b (+ a b)]);; new prev is old curr, new curr is sum of both previous numbers[01])));; recursive base case: prev 0, curr 1

A more elegant solution is inspired by the Haskell implementation of an infinite list of Fibonacci numbers:

This can be improved to an O(n) solution, like the iterative solution, by memoizing the function so that numbers that have been computed are cached. Like a lazy sequence, this also has the advantage that subsequent calls to the function use previously cached results rather than recalculating.

To find the n{\displaystyle n}th Fibonacci number, set the initial value of count equal to n{\displaystyle n}–2 and run the program. The machine will halt with the answer stored in the accumulator. Since Computer/zero's word length is only eight bits, the program will not work with values of n{\displaystyle n} greater than 13.

Here are four versions of Fibonacci Number calculating functions. FibD has an argument limit of magnitude 84 due to floating point precision, the others have a limit of 92 due to overflow (long).The traditional recursive version is inefficient. It is optimized by supplying a static storage to store intermediate results. A Fibonacci Number generating function is added.
All functions have support for negative arguments.

Since there are only a fixed and small amount of Fibonacci numbers that fit in a machine word, this FORTH version creates a table of Fibonacci numbers at compile time. It stops compiling numbers when there is arithmetic overflow (the number turns negative, indicating overflow.)

Approach similar to the Matrix algorithm in C#, with some shortcuts involved.
Since it uses exponentiation by squaring, calculations of fib(n) where n is a power of 2 are particularly quick.
Eg. fib(2^20) was calculated in a little over 4 seconds on this poster's laptop.

fib 0=0-- this line is necessary because "something ^ 0" returns "fromInteger 1", which unfortunately-- in our case is not our multiplicative identity (the identity matrix) but just a 1x1 matrix of 1fib n =last$head$ unMat $(Mat [[1,1],[1,0]])^ n

So, for example, the hundred-thousandth Fibonacci number starts with the digits

Now, an O(logN) solution.
For large N, it takes far longer to convert the result to a string for output
than to do the actual computation.
This example computes fib(1000000) if there is no integer argument.

jq does not (yet) have infinite-precision integer arithmetic, and
currently the following algorithms only give exact answers up to fib(78). At a certain point, integers are converted to floats,
but floating point precision for fib(n) fails after n = 1476:
in jq, fib(1476) evaluates to 1.3069892237633987e+308

Recent versions of jq (after July 1, 2014) include basic optimizations for tail recursion, and nth_fib is defined here to take advantage of TCO. For example, nth_fib(10000000) completes with only 380KB (that's K) of memory. However nth_fib can also be used with earlier versions of jq.

The following is a jq generator which produces the first n terms of the Fibonacci sequence efficiently, one by one. Notice that it is simply a variant of the above tail-recursive function. The function is in effect turned into a generator by changing "( _ | fib )" to "$sum, (_ | fib)".

This image is a VI Snippet, an executable image of LabVIEW code. The LabVIEW version is shown on the top-right hand corner. You can download it, then drag-and-drop it onto the LabVIEW block diagram from a file browser, and it will appear as runnable, editable code.

The above implementations may be too simplistic, as the first is incredibly slow for any reasonable range due to nested recursions and while the second is faster it uses an increasing amount of memory. The following uses recursion much more effectively while not using memory:

However, the recursive approaches in Mathematica are limited by the limit set for recursion depth (default 1024 or 4096 for the above cases), limiting the range for 'n' to about 1000 or 2000. The following using an iterative approach has an extremely high limit (greater than a million):

The MATLAB help file suggests an interesting method of generating the Fibonacci numbers. Apparently the determinate of the Dramadah Matrix of type 3 (MATLAB designation) and size n-by-n is the nth Fibonacci number. This method is implimented below.

Mercury is both a logic language and a functional language. As such there are two possible interfaces for calculating a Fibonacci number. This code shows both styles. Note that much of the code here is ceremony put in place to have this be something which can actually compile. The actual Fibonacci number generation is contained in the predicate fib/2 and in the function fib/1. The predicate main/2 illustrates first the unification semantics of the predicate form and the function call semantics of the function form.

The provided code uses a very naive form of generating a Fibonacci number. A more realistic implementation would use memoization to cache previous results, exchanging time for space. Also, in the case of supplying both a function implementation and a predicate implementation, one of the two would be implemented in terms of the other. Examples of this are given as comments below.

But what if you want the speed of the fib_acc with the recursive (more declarative) definition of fib? Then use memoization, because Mercury is a pure language fib(N, F) will always give the same F for the same N, guaranteed. Therefore memoization asks the compiler to use a table to remember the value for F for any N, and it's a one line change:

We've shown the definition of fib/2 again, but the only change here is the memoization pragma (see the reference manual). This is not part of the language specification and different Mercury implementations are allowed to ignore it, however there is only one implementation so in practice memoization is fully supported.

Memoization trades speed for space, a table of results is constructed and kept in memory. So this version of fib consumes more memory than than fib_acc. It is also slightly slower than fib_acc since it must manage its table of results but it is much much faster than without memoization. Memoization works very well for the Fibonacci sequence because in the naive version the same results are calculated over and over again.

FIBOITER(N) ;Iterative version to get the Nth Fibonacci number ;N must be a positive integer ;F is the tree containing the values ;I is a loop variable. QUIT:(N\1'=N)!(N<0) "Error: "_N_" is not a positive integer." NEW F,I SET F(0)=0,F(1)=1 QUIT:N<2 F(N) FOR I=2:1:N SET F(I)=F(I-1)+F(I-2) QUIT F(N)

The previous way is the naive form, because for most n the fib_rec is called twice, and it is not tail recursive because it adds the result of two function calls. The next version resolves these problems through accumulator argument technique. It is computationally equivalent to the iterative version above (tail recursion is effectively iteration):

(* support for negatives *)let fib n =if n <0&& n mod2=0then minus_num (fib (abs n))else fib (abs n);;(* It can be called from the command line with an argument *)(* Result is send to standart output *)let n =int_of_stringSys.argv.(1)inprint_endline(string_of_num (fib n))

This is an exact version of the above formula. quadgen(5) represents ϕ{\displaystyle \phi } and the number is stored in the form a+bϕ{\displaystyle a+b\phi }. imag takes the coefficient of ϕ{\displaystyle \phi }. This uses the relation

ϕn=Fn−1+Fnϕ{\displaystyle \phi ^{n}=F_{n-1}+F_{n}\phi }

and hence real(quadgen(5)^n) would give the (n-1)-th Fibonacci number.

fib(n)=imag(quadgen(5)^n)

A more direct translation (note that 5=2ϕ−1{\displaystyle {\sqrt {5}}=2\phi -1}) would be

All n×n (0,1) lower Hessenberg matrices have determinant at most F(n). The n×n anti-Hadamard matrix[1] matches this upper bound, and hence can be used as an inefficient method for computing Fibonacci numbers of positive index. These matrices are the same as Matlab's type-3 "Dramadah" matrices, following a naming suggestion of C. L. Mallows according to Graham & Sloane.

Iterative and tail recursive solutions were only accurate to 43, and a standard recursive version was way too slow above 35, so I memoised it. Using native integers/atoms, errors creep in above 78

sequence fcache = {1,1}

function fibonamem(integer n) -- memoized, works for -ve numbers, inaccurate above 78integer absn = abs(n) if n=0 then return 0 end if if absn>length(fcache) then fcache = append(fcache,fibonamem(absn-1)+fibonamem(absn-2)) if absn!=length(fcache) then ?9/0 end if end if if n<0 and remainder(n,2)=0 then return -fcache[absn] end if return fcache[absn]end function

for i=0 to 30 do printf(1,"%d", fibonamem(i)) if i!=30 then puts(1,", ") end ifend forputs(1,"\n")

function fibonamemba(integer n) -- memoized, works for -ve numbers, yields bigatominteger absn = abs(n) if n=0 then return BA_ZERO end if if absn>length(fcacheba) then fcacheba = append(fcacheba,ba_add(fibonamemba(absn-1),fibonamemba(absn-2))) if absn!=length(fcacheba) then ?9/0 end if end if if n<0 and remainder(n,2)=0 then return ba_sub(0,fcacheba[absn]) end if return fcacheba[absn]end function

for i=0 to 30 do ba_printf(1,"%B", fibonamemba(i)) if i!=30 then puts(1,", ") end ifend forputs(1,"\n")ba_printf(1,"%B", fibonamemba(777))puts(1,"\n")

There seems to be a limitation (dare I say, bug?) in PowerBASIC regarding how large numbers are stored. 10E17 and larger get rounded to the nearest 10. For F(n), where ABS(n) > 87, is affected like this:

In this case by asserting the new N,Value pairing as a rule in the database we're making the classic time/space tradeoff. Since the space costs are (roughly) linear by N and the time costs are exponential by N, the trade-off is desirable. You can see the poor man's memoizing easily:

One of my favorites; loosely similar to the first example, but without the performance penalty, and needs nothing special to implement. Not even a dynamic database predicate. Attributed to M.E. for the moment, but simply because I didn't bother to search for the many people who probably did it like this long before I did. If someone knows who came up with it first, please let us know.

What I really like about this one, is it is also a generator- i.e. capable of generating all the numbers in sequence needing no bound input variables or special Prolog predicate support (such as freeze/3 in the previous example):

The recursive code as written two sections above is incredibly slow and inefficient due to the nested recursion calls. Although the memoization above makes the code run faster, it is at the cost of extra memory use. The below code is syntactically recursive but actually encodes the efficient iterative process, and thus doesn't require memoization:

However, although much faster and not requiring memory, the above code can only work to a limited 'n' due to the limit on stack recursion depth by Python; it is better to use the iterative code above or the generative one below.

# To understand why this matrix is useful for Fibonacci numbers, remember# that the definition of Matrix.**2 for any Matrix[[a, b], [c, d]] is# is [[a*a + b*c, a*b + b*d], [c*a + d*b, c*b + d*d]]. In other words, the# lower right element is computing F(k - 2) + F(k - 1) every time M is multiplied# by itself (it is perhaps easier to understand this by computing M**2, 3, etc, and# watching the result march up the sequence of Fibonacci numbers).

M = Matrix[[0, 1], [1,1]]

# Matrix exponentiation algorithm to compute Fibonacci numbers.# Let M be Matrix [[0, 1], [1, 1]]. Then, the lower right element of M**k is# F(k + 1). In other words, the lower right element of M is F(2) which is 1, and the# lower right element of M**2 is F(3) which is 2, and the lower right element# of M**3 is F(4) which is 3, etc.## This is a good way to compute F(n) because the Ruby implementation of Matrix.**(n)# uses O(log n) rather than O(n) matrix multiplications. It works by squaring squares# ((m**2)**2)... as far as possible# and then multiplying that by by M**(the remaining number of times). E.g., to compute# M**19, compute partial = ((M**2)**2) = M**16, and then compute partial*(M**3) = M**19.# That's only 5 matrix multiplications of M to compute M*19. defself.fib_matrix(n)return0if n <= 0# F(0)return1if n == 1# F(1)# To get F(n >= 2), compute M**(n - 1) and extract the lower right element.return CS::lower_right(M**(n -1))end

"Fibers are primitives for implementing light weight cooperative concurrency in Ruby. Basically they are a means of creating code blocks that can be paused and resumed, much like threads. The main difference is that they are never preempted and that the scheduling must be done by the programmer and not the VM." [2]

This uses a feature from nightly Rust which makes it possible to (cleanly) return an iterator without the additional overhead of putting it on the heap. In stable Rust, we'd need to return a Box<Iterator<Item=f64>> which has the cost of an additional allocation and the overhead of dynamic dispatch. The version below does not require the use of the heap and is done entirely through static dispatch.

Although the tail recursive version above is quite efficient, it only generates the final nth Fibonacci number and not the sequence up to that number without wasteful repeated calls to the procedure/function.

The following procedure generates the sequence of Fibonacci numbers using a simplified version of a lazy list/stream - since no memoization is requried, it just implements future values by using a zero parameter lambda "thunk" with a closure containing the last and the pre-calculated next value of the sequence; in this way it uses almost no memory during the sequence generation other than as required for the last and the next values of the sequence (note that the test procedure does not generate a linked list to contain the elements of the sequence to show, but rather displays each one by one in sequence):

# First we need to convert each number into the right number of ticks # Start by marking digitss/[0-9]/<&/g

# We have to do the digits manually.s/0//g; s/1/|/g; s/2/||/g; s/3/|||/g; s/4/||||/g; s/5/|||||/gs/6/||||||/g; s/7/|||||||/g; s/8/||||||||/g; s/9/|||||||||/g

# Multiply by ten for each digit from the front.:tenss/|</<||||||||||/gt tens

# Done with digit markerss/<//g

# Now the actual work.:split# Convert each stretch of n >= 2 ticks into two of n-1, with a mark betweens/|\(|\+\)/\1-\1/g# Convert the previous mark and the first tick after it to a different mark# giving us n-1+n-2 marks.s/-|/+/g# Jump back unless we're done.t split# Get rid of the pluses, we're done with them.s/+//g

const func integer: fib (in integer: number) is func result var integer: result is 1; begin if number > 2 then result := fib(pred(number)) + fib(number - 2); elsif number = 0 then result := 0; end if; end func;

const func bigInteger: fib (in integer: number) is func result var bigInteger: result is 1_; local var integer: i is 0; var bigInteger: a is 0_; var bigInteger: c is 0_; begin for i range 1 to pred(number) do c := a; a := result; result +:= c; end for; end func;

SkookumScript is designed to work in tandem with C++ and its strength is at the high-level stage-direction of things. So when confronted with benchmarking scripting systems it is genrally better to make a built-in call. Though in most practical cases this isn't necessary.

CREATEFUNCTION fib(n int)RETURNSnumericAS $$-- This recursive with generates endless list of Fibonacci numbers.WITHRECURSIVE fibonacci(current, previous)AS(-- Initialize the current with 0, so the first value will be 0.-- The previous value is set to 1, because its only goal is not-- special casing the zero case, and providing 1 as the second-- number in the sequence.---- The numbers end with dots to make them numeric type in-- Postgres. Numeric type has almost arbitrary precision-- (technically just 131,072 digits, but that's good enough for-- most purposes, including calculating huge Fibonacci numbers)SELECT0., 1.UNIONALL-- To generate Fibonacci number, we need to add together two-- previous Fibonacci numbers. Current number is saved in order-- to be accessed in the next iteration of recursive function.SELECT previous + current, currentFROM fibonacci)-- The user is only interested in current number, not previous.SELECTcurrentFROM fibonacci-- We only need one number, so limit to 1LIMIT1-- Offset the query by the requested argument to get the correct-- position in the list.OFFSET n$$ LANGUAGE SQL RETURNSNULLONNULLINPUTIMMUTABLE;

Calculates the tenth Fibonacci number. To calculate the nth, change the initial value of the counter to n-1 (subject to the restriction that the answer must be small enough to fit in a signed 32-bit integer, the SSEM's only data type). The algorithm is basically straightforward, but the absence of an Add instruction makes the implementation a little more complicated than it would otherwise be.

In Tcl 8.6 a tailcall function is available to permit writing tail-recursive functions in Tcl. This makes deeply recursive functions practical. The availability of large integers also means no truncation of larger numbers.

This example is incorrect. Please fix the code and remove this message.

Details: There is a race between parallel commands. tee last might open and truncate the file before cat last opens it. Then cat last pipes the empty file to xargs, and expr reports a syntax error, and the script hangs forever.

The analytical method uses arbitrary precision floating point
arithmetic from the mpfr library and then converts the result to
a natural number. Sufficient precision for an exact result is
always chosen based on the argument. This test program computes the first
twenty Fibonacci numbers by all three methods.